Solid convergence spaces

M. Schroder
1973 Bulletin of the Australian Mathematical Society  
The category of solid convergence spaces is introduced, and shown to l i e s t r i c t l y between the category of a l l convergence spaces and that of pseudo-topological spaces. A wide class of convergence spaces, including the e-embedded spaces of Binz, is then characterized in terms of this concept. Finally, several i l l u s t r a t i v e examples are given. The categories L of convergence spaces, P of principal convergence spaces (also known as pseudo-topological spaces) and T of
more » ... and T of topological spaces are a l l more or less familiar. It is well known that any topological space is completely determined by the collection of a l l i t s open covers. After having introduced an analogous concept for convergence spaces (namely, indexed cover), we find instead that only some convergence spaces can be so determined. Such spaces are called solid. Some properties of solid spaces are listed, and i t is shown that the inclusions P c 5 c i. are both proper, where S denotes the category of solid spaces. In addition, using these ideas we discuss in more detail a more restricted class of spaces, obtaining as a special case the internal characterization of e-embedded convergence spaces given independently by Muller [6]. This states that a space is e-embedded iff i t is Hausdorff, solid and u-regular (this last term generalizing complete regularity for topological spaces). Finally, examples are given showing among other things that these three conditions are independent of one another.
doi:10.1017/s0004972700042738 fatcat:pfz7sxqprjar3p6qgyeqb3hc74